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In the mathematical field of order theory, an element ''a'' of a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
with least element 0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0.


Atomic orderings

Let <: denote the
covering relation In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically expr ...
in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element ''b'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite partially ordered set with 0 is atomic, but the set of nonnegative real numbers (ordered in the usual way) is not atomic (and in fact has no atoms). A partially ordered set is relatively atomic (or ''strongly atomic'') if for all ''a'' < ''b'' there is an element ''c'' such that ''a'' <: ''c'' ≤ ''b'' or, equivalently, if every interval 'a'', ''b''is atomic. Every relatively atomic partially ordered set with a least element is atomic. Every finite poset is relatively atomic. A partially ordered set with least element 0 is called atomistic (not to be confused with atomic) if every element is the least upper bound of a set of atoms. The linear order with three elements is not atomistic (see Fig. 2). Atoms in partially ordered sets are abstract generalizations of
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
s in set theory (see Fig. 1). Atomicity (the property of being atomic) provides an abstract generalization in the context of order theory of the ability to select an element from a non-empty set.


Coatoms

The terms ''coatom'', ''coatomic'', and ''coatomistic'' are defined dually. Thus, in a partially ordered set with greatest element 1, one says that * a coatom is an element covered by 1, * the set is coatomic if every ''b'' < 1 has a coatom ''c'' above it, and * the set is coatomistic if every element is the greatest lower bound of a set of coatoms.


References

*


External links

* * {{planetmath reference, urlname=Poset, title=Poset Order theory